A note on matrix equation and its control application

In this paper, by means of real representation of a quaternion matrix, the quaternion matrix equation R(X) = E is studied. The explicit solution algorithm and iterative solution algorithm are both given. By the iterative method, the solvability of the quaternion matrix equation R(X) = E can be determined automatically. When the quaternion matrix equation R(X) = E is consistent, a solution can be obtained within finite iterative steps in the absence of roundoff errors for any initial quaternion matrix X0. Meanwhile, if a special kind of initial quaternion matrix is chosen, the least norm solution can also be obtained. Finally, both numerical example and control application example are given to show the efficiency of the presented iterative methods.